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Direct CP violation in $K\to ππ$ decays and supersymmetry

The quantities $ε_K^\prime$ and $ε_K$ measure the amount of direct and indirect CP violation in $K\to ππ$ decays, respectively. Using the recent lattice results from the RBC and UKQCD Collaborations and a new compact implementation of the $ΔS=1$ renormalization group evolution we predict $ \mbox{Re}\, \frac{ε_{K}'}{ε_{K}} = \left(1.06 \pm 5.07 \right) \times 10^{-4}$ in the Standard Model. This value is $2.8\,σ$ below the experimental value of $ \mbox{Re}\, \frac{ε_{K}'}{ε_{K}} = \left(16.6 \pm 2.3 \right) \times 10^{-4}.$ In generic models of new physics the well-understood $ε_K$ precludes large contributions to $ε_K^\prime$, if the new contributions enter at loop level. However, one can resolve the tension in $ε_{K}'/ε_{K}$ within the Minimal Supersymmetric Standard Model. To this end two features of supersymmetry are crucial: First, one can have large isospin-breaking contributions (involving the strong instead of the weak interaction) which enhance $ε_K^\prime$. Second, the Majorana nature of gluinos permits a suppression of the MSSM contribution to $ε_K$, because two box diagrams interfere destructively.